Subgroups — #14517258

Order 1944.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	#10619575	C₃	1	9	yes	maximal, complemented
4	C₂×#6911	—	4	1	no	maximal
5	C₃×C₄×3₋¹⁺²	—	3	1	no	—
6	#6911	—	4	1	no	Sylow
7	C₃³×Q₈	C₃²	1	1	yes	characteristic
8	C₃²×SL(2,𝔽₃)	—	3	3	no	—
9	Q₈×3₋¹⁺²	—	3	3	no	—
10	C₃×Q₈×C₉	C₃²	1	3	yes	complemented
11	Q₈⋊H(3,𝔽₃)	—	3	6	no	—
12	Q₈⋊H(3,𝔽₃)	C₃²	1	9	yes	complemented
13	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
14	C₂×C₃×H(3,𝔽₃)	—	4	3	no	—
15	C₂×#732	—	4	9	no	—
16	C₃³×C₄	—	3	1	no	—
17	C₄×3₋¹⁺²	—	3	9	no	—
18	C₃×C₄×C₉	—	3	3	no	—
19	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
20	C₃×H(3,𝔽₃)	—	4	3	no	—
21	#732	—	4	9	no	—
22	C₃²×Q₈	—	9	1	no	—
23	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
24	C₃²×Q₈	C₃³	1	1	yes	characteristic
25	C₃×SL(2,𝔽₃)	—	3	9	no	—
26	C₃×SL(2,𝔽₃)	—	9	9	no	—
27	Q₈×C₉	—	3	3	no	—
28	C₂×C₃³	—	12	3	no	—
29	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
30	C₂×3₋¹⁺²	—	3	3	no	—
31	C₂×H(3,𝔽₃)	—	4	9	no	—
32	C₂×H(3,𝔽₃)	—	12	6	no	—
33	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
34	C₃²×C₄	—	3	1	no	—
35	C₃²×C₄	—	3	3	no	—
36	C₃²×C₄	—	9	3	no	—
37	C₄×C₉	—	3	9	no	—
38	C₃³	—	12	3	no	—
39	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
40	3₋¹⁺²	—	3	3	no	—
41	H(3,𝔽₃)	—	4	9	no	—
42	H(3,𝔽₃)	—	12	6	no	—
43	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	complemented
44	C₃×Q₈	—	3	1	no	—
45	C₃×Q₈	—	3	3	no	—
46	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	SL(2,𝔽₃)	—	9	9	no	—
48	C₂×C₃²	—	9	1	no	—
49	C₂×C₃²	—	12	9	no	—
50	C₂×C₃²	—	36	9	no	—
51	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
52	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
53	C₂×C₉	—	3	3	no	—
54	C₃×C₄	—	3	3	no	—
55	C₃×C₄	—	3	1	no	—
56	C₃×C₄	—	9	3	no	—
57	C₃²	—	9	1	no	—
58	C₃²	—	12	9	no	—
59	C₃²	—	36	9	no	—
60	C₃²	Q₈⋊H(3,𝔽₃)	1	3	yes	—
61	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
62	C₉	—	3	3	no	—
63	Q₈	#6911	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	3	1	no	—
65	C₂×C₃	—	3	3	no	—
66	C₂×C₃	—	36	9	no	—
67	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
68	C₄	—	3	1	no	—
69	C₃	—	3	3	no	—
70	C₃	—	3	1	no	—
71	C₃	—	36	9	no	—
72	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
73	C₂	#11757852	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	C₃×Q₈×3₋¹⁺²	C₃
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6911
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃³ ◁ C₂×C₃×3₋¹⁺²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×Q₈×3₋¹⁺²	C₃	semidirect
2. C₃×Q₈⋊H(3,𝔽₃)	C₃	semidirect
3. #10619575	C₃	semidirect
10. C₃×Q₈×C₉	C₃²	semidirect
12. Q₈⋊H(3,𝔽₃)	C₃²	semidirect
19. C₃×3₋¹⁺²	SL(2,𝔽₃)	semidirect
43. C₃×C₉	C₃×SL(2,𝔽₃)	semidirect
63. Q₈	#6911	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×Q₈×3₋¹⁺²	3
C₃×Q₈⋊H(3,𝔽₃)	3
#10619575	3
C₂×#6911	4